This paper investigates the prescribed-time fully distributed Nash equilibrium seeking (PT-FDNES) problem for nonlinear multi-agent systems (MASs) over weight-unbalanced directed graphs (digraphs). To counterbalance unweighted communication flows caused by the unbalanced network, the temporal transformation technique is first introduced to derive a consensus-based algorithm for calculating the left eigenvector of the Laplacian matrix, based on which a prescribed-time fully distributed estimator with adaptive gains is developed to obtain the NE without requiring any global information. By utilizing this NE estimator to formulate an intermediate control law, the NE seeking problem is then effectively transformed into a local reference-tracking problem. To deal with the non-differentiability issue caused by the NE estimator and ensure the exact tracking, the dynamic gain technique and novel scaling functions are employed to derive a prescribed-time tracking controller which completely compensates for the mismatched uncertainties while reducing the computational burden. The proposed control framework enables the actions of all agents to reach the NE within a prescribed time. In addition, the obtained results are further re-designed to exhibit the robustness by introducing an improved -modification scheme without sacrificing the convergence performance. Numerical examples are provided to demonstrate the validity of the theoretical results.

中文翻译：

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非平衡有向图上非线性多智能体系统的规定时间全分布纳什均衡求


本文研究了权重不平衡有向图（有向图）上的非线性多智能体系统（MAS）的规定时间完全分布式纳什均衡寻求（PT-FDNES）问题。为了抵消不平衡网络造成的未加权通信流，首先引入时间变换技术，推导出基于共识的拉普拉斯矩阵左特征向量计算算法，并在此基础上开发了具有自适应增益的规定时间全分布式估计器无需任何全局信息即可获取 NE。通过利用该NE估计器来制定中间控制律，NE搜索问题可以有效地转化为局部参考跟踪问题。为了解决NE估计器引起的不可微性问题并确保精确跟踪，采用动态增益技术和新颖的缩放函数来推导规定时间跟踪控制器，该控制器完全补偿失配的不确定性，同时减少计算负担。所提出的控制框架使得所有代理的动作能够在规定的时间内到达NE。此外，通过引入改进的修改方案，在不牺牲收敛性能的情况下，进一步重新设计所获得的结果，以展现鲁棒性。提供数值例子来证明理论结果的有效性。